Optimal bounds in Taylor--Couette flow

In this presentation, we find optimal bounds on mean quantities, such as the energy dissipation rate, torque and the Nusselt number in Taylor--Couette flow using the well-known background method. The main result of our study is that we can obtain the analytical expression of the dependence of these optimal bounds on the radius ratio, which is the geometrical parameter in the problem. First, we find the optimal bounds by numerically solving the Euler--Lagrange equations obtained from a variational formulation of the background method. We then obtain suboptimal but analytical bounds on the mean quantities using analysis techniques that employ a definition of the background flow with two boundary layers (near the inner and the outer cylinder), whose relative thicknesses were optimized to obtain this suboptimal bound. Crucially, the optimal bounds have the same dependence on the radius ratio as the suboptimal bounds in the limit of high Reynolds number. We compare the geometrical dependence of optimal bounds on mean quantities with the geometrical dependence observed in the DNS and experiments of turbulent Taylor–Couette flow. Further, in this study, we probed into the structure of the optimal perturbed flow in the Taylor--Couette problem and using the insights gained here, we rigorously dismiss the applicability of the background method to certain flow problems.